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The wind-up/shake flashlights I'm familiar with don't use batteries (who wants to crank/shake for four hours?) They use monster capacitors (i.e. one farad - yes, someone actually came up with a practical use for those nerd cat-toys). If you need more (ahem) capacity, you could theoretically add more capacitors. The only electrical question I have is whether the dynamo can handle the additional current. My offhand guess is that it probably will. It's easy enough to test. Charge and discharge several times, then check the temperature of the dynamo. If it's hot, charge the caps one at a time.
What concerns me more is the additional wear on the dynamo (charging more caps will require harder cranking), the cheap plastic gears and housing, and the lack of bearings. Being cheap and lazy, I'd probably go with 15 minute NiMH's.
I got two similar flashlights free for trading burnt out light bulbs for re-cycling.
One lighted for more than 24 hours but was not very bright and during the test it kept dimming. After about 2 hours it was so dim that it was useless as a flashlight.
Inside there is a plastic box around a very small super-cap so I couldn't see it well.
My arm wore out cranking it. I think its cheap plastic gears will fail soon.
I read an interesting suggestion, using a fly wheel and a pull string to run a dynamo. It works like a rope for starting a lawn mower. This theoretically could generate a lot more electricity with a much more natural motion.
Using Schottky diodes for rectification (assuming it's a rotating permanent magnet inside a stator coil) rather than the cheap 50 cents per dustpanful bipolar silicon 1N4001's they probably use in these lights would increase efficiency by minimizing forward voltage drop.
Normal silicon diodes have around 0.6 to 1.7 volts forward voltage drop. Schottky diodes (AKA hot carrier diodes) have a much lower forward voltage drop, around 0.15 volts, which is even better than germanium diodes.
The "cats whisker" crystal detector in crystal radios is a point contact rectifier, a primitive form of Schottky diode. Cat's-whisker detector - Wikipedia, the free encyclopedia I've built quite a few crystal radios, partly for fun, but largely because back then we were dirt poor Cajuns and that's all I could afford.
I have two identical shake flashlights I got at an open air market.
One of them has a magnet that slides inside a coil, which feeds a standard small diode and a tiny 25mAh NiCd battery. I can shake it and use it any time I need it.
The other one looks like it has a magnet that slides inside the same coil. But the coil wires aren't connected to anything and the "magnet" is just plain iron. It has a CR2016 cell instead of the NiCd battery. I didn't have to shake it, but it only worked for a few hours.
The boxes they came in look the same, even the same words are spelled wrong. The flashlights look the same unless you really look closely.